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On a set S, we say that $\leq$ is a partial ordering, if for $s_1,s_2,s_3 \in S$
  1. $s_1\leq s_2, s_2 \leq s_1 \Longrightarrow s_1=s_2$
  2. $s_1 \leq s_2 \leq s_3 \Longrightarrow s_1 \leq s_3$
  3. $s_1 \leq s_1$